Bring the Tools, not the Weapons
The hard part here is getting the precision tooling you will need done right. Rifling a barrel and pressing ammunition is pretty darn hard without the right equipment, but all the tools needed to set up a fire-arms and munitions workshop take up surprisingly little space.
Since he has enough wealth in the 21st century to acquire significant amounts of gold and build a time-machine, I'm assuming he's fairly wealthy. Although he lacks mechanical engineering skills himself, he could prepare for his mission by hiring proper engineers to modify hobbyist gun making kits to rely on non-electrical power sources such as foot-pedals or a water wheel, and help him design weapons to the specifications of the alloys that will be available to him in the past.
Before he goes back, he should then run tests making his own home-made bronze, gun powder, ignition caps, etc. to test his ability to make sound weapons from what will be local resources.
The Time & Place
For this I'd suggest going to Ancient Athens at ~600 BCE.
While most historical guns were made of cast iron or steel, bronze is actually a fairly good material for making firearms, because it can be caste without being brittle. A revolutionary era bronze cannon was half the weight of the cast iron cannons of the time. Steel did not really get good enough to outperform bronze as a gun-making material until the 1800s, so your time traveler should head back to a civilization that is proficient in bronze working before gunpowder was first invented. This means that in the you will have a good material to work with and be able to control the means of producing and distributing the ammo so that even if another nation reverse engineers your gun, they won't have the means to make the ammo.
While 600 BCE was technically after the Bronze Age collapse, bronze was still widely used at this time since Athens re-established the trade routes that made bronze possible again. While bronze was not their preferred alloy in weapon/armor making as it was in the Mycenean Period, it was still prominently used in their artwork; so, there still would have been a large number of skilled bronze craftsmen.
The biggest reason to pick Athens though over an older, true bronze age civilization is that bronze age civilizations were pretty exclusively ruled by nobilities or theocracies such that you could never gain a position of power among the people as an outsider. No matter how good your tech is, you would never be more than the ruler's slave. In Ancient Greece, money could buy political power. You could raise your self up to be a general, and use the influence of your conquests to eventually usurp or at least strongly control the democracy.
Athens also has a relatively large population in a small place with lots of allied city states. This gives you the manpower you need to expand far.
The Currency
You may want to exchange most of your gold for silver before you go. At today's market value, gold has an 80:1 exchange rate with silver, but in ancient Greece, the exchange rate would be anywhere from about 13:1 to 1:3 just depending. Silver (and not gold) was the standard currency in ancient Athens and the preferred coinage for international trade on the Mediterranean (4.3g was worth 1 day's labor). Greeks specifically did not like being payed in gold due to efforts by Persia to devalue their silver coinage by intentionally fixing the exchange rate of gold and silver to prefer their coinage at the 13:1 ratio, even though gold was considered less valuable than silver in Greece. So, definitely bring some gold for any materials you need to buy from Persian merchants, but silver is the way to go for local labor and materials.
The Weapon he Makes
To dominate the ancient world, your weapon needs a few qualities that not all historical firearms could achieve. While older firearms like culivans, muskets, and arquebuses offer some specific advantages over bows and crossbows that made them successful in their own hay day, their advantages are not significant enough to really make them "world domination" weapons. Instead, your minimum firearm should be something that is so much better than older weapons that even archery cultures like the Ancient Cretans, Persians, or Parthians would be at a distinct disadvantage since these cultures could arguably out range war most early gunpowder armies.
First, it will need to be able to kill through the shields and armor of the time period you go back to. This significantly limits the effectiveness of most shotguns and low caliber firearms since available resources will likely not be able to achieve the muzzle velocity and armor penetration of modern munitions. Before the invention of more complex gunpowders like guncotton, nearly all firearms were .45cal or larger for this reason.
Secondly, it will need range. To have a true tactical advantage, you will need to be able to shoot down enemy archers, slingers, and crossbowmen before they can start shooting at you. Most muskets are only target accurate at 100m and area accurate up to 300m (similar to ancient bows). More primitive firearms like early culverins and fire lances were only target accurate within a few paces making them no more accurate than an untrained archer. Even though they had a theoretical range of 600m they were not even considered area accurate at these sorts of distances. The feature that really made guns hands-down better than of bows and crossbows was the invention of riffling. Even primitive rifles could give you target accuracy in the 700-1800m range.
Third, you will need a enough rate of fire that your armies will get in many shots before your enemies can close range enough to take away your advantages. Automatic firing systems might be too complicated to reproduce since you may not be able to produce high precision parts, but revolvers can be made to relatively low tolerances. Also, if individual bullets prove to be too much work to make, a removable revolver cylinder like that found on the Colt Paterson would allow one to pre-pack cylinders to more or less the same effect as having clips of bullets.
The hardest part is going to be primer caps. While your time traveler may struggle to make these himself, all the chemicals required to make a primitive Mercury Fulminate (mercury, potassium nitrate, green vitriol, and ethanol) have all been known about and used since ancient times; so, an ancient apothecary with a little bit of guidance could probably solve this problem. If not, a sort of flint-lock mechanism could be employed, though it may be a bit less convenient/reliable.
The final weapon will likely resemble the Whitworth Rifle as suggests UIDAlexD in terms of accuracy, calbre, and muzzle velocity. The addition of a revolver cylinder may slightly reduce the weapon's accuracy, but you'd still be way outranging bows, and the increased rate of fire would more than be worth it.